State Estimation in Linear and Non-Linear Systems

in reference to Chemical Engineering

Submitted in partial fulfillment of the requirements of the degree of

Bachelor of Technology

by :

Mayank Sanjay Kumar Gupta - 211916

Wakde Vaibhav Mohanrao - 211927

Supervisor :

Dr. Venkata Goutham Polisetty

Department of Chemical Engineering

NATIONAL INSTITUTE OF TECHNOLOGY ANDHRA PRADESH

2022-2023

Table of Contents

Sr.no

Title

Page Number

Introduction

Review of Literature

Report on the Present Investigation

3.1

Theoretical and Mathematical Derivation

3.1.1

Bayes Filter

3.1.2

Kalman Filter

Results and Discussion

Summary and Conclusions

Appendix

6.1

Appendix-1

6.2

Appendix-2

Literature Cited

Chapter 1

Introduction

Kalman Filter is a linear quadratic estimation (LQE) algorithm ,which utilizes a series of

measurements over interval of time , along with the statistical noises , to estimate the value of a

variable .We do not obtain accurate results with single set of measurements hence Kalman Filter

(KF) makes use of a joint probability distribution over the variables for each timeframe.

Kalman Filter has wide applications as it is an optimal estimator , but it is valid under

certain assumptions -

● Errors have a normal (Gaussian) distribution.

● Physical random phenomena due to primary random sources exciting dynamic

systems.Physical random phenomena may be thought of as due to primary random

sources exciting dynamic systems

● The primary sources are assumed to be independent gaussian random processes with zero

mean; the dynamic systems will be linear. [1]

Kalman Filter is a linear estimator but it may be extended to non linear systems by means

of - EKF (Extended Kalman Filter) and

UKF (Unscented Kalman Filter) .Kalman

Filter is based on Markov's model.KF can

be understood by help of Figure-1.1- [1]

and has Prediction and updation step.

Figure-1.1 - Kalman Filter

Kalman filtering has numerous technological applications. A common application is for

guidance, navigation, and control of vehicles, particularly aircraft, spacecraft and ships

positioned dynamically.

Through this project we aim to estimate the state of linear and nonlinear systems with

respect to chemical engineering .In the initial part of the project we go over the theoretical and

mathematical conceptualization of the Kalman Filter .

Then we try to model and simulate the given example with the help of a Python

code. Similarly KF can be applied to a CSTR to estimate the parameters such as concentration or

pressure.The non linear filters may be used for real-time implementable optimization of the Fuel

Efficiency of Solid Oxide Fuel Cells.

Chapter 2

Review of Literature

Kalman Filter has been an area of research for several decades and various papers have

been published on it. While working on this project we referred to - Constrained Nonlinear State

Estimation Using Ensemble Kalman Filters [2] and Constrained nonlinear state estimation based

on the UKF approach [3] .

The key takeaways from the first paper [2] were :

● Proposed a constrained recursive formulation of the ensemble Kalman filter (EnKF) that

retains the advantages of unconstrained EnKF while systematically dealing with bounds

on the estimated state variables.

● The performance of the proposed constrained EnKF is compared with the performances

obtained using the recursive constrained formulations available in the literature using two

benchmark examples from the literature (a gas-phase reactor and an isothermal batch

reactor), which involve constraints on the estimated state variables.

● When compared, the performance of the proposed C-EnKF formulation was found to be

significantly better than those obtained using RNDDR and C-UKF formulations.

Furthermore, the performance of the proposed C-EnKF scheme was found to be

satisfactory when employed for state estimation in a system having constraints on process

noise.

The key takeaways from the second paper [3] were :

● Overview of several UKF based nonlinear estimation algorithms as an alternative to the

EKF

● Suggested a reformulation of the correction step which can be applied to all of the

presented UKF algorithms, presented a QP formulation of the NLP UKF (which also can

be applied to all of the presented UKF algorithms)

● Proposed alternatives to realize constraints within the UKF approach

Chapter 3

Report on the Present Investigation

3.1 Theoretical and Mathematical Derivation

3.1.1 Bayes Filter

The Kalman Filter is based on the Bayes theorem and Markov Assumption. From the

Bayes theorem we get the following equation :

The above equation shows the probability of event A

occuring when the event B has

occurred.The above result then may be extended to multiple events whose detailed derivation has

been given in Appendix-1.

On the other hand the Markov Assumption may be understood using Figure 3.1

Here

We see that at time t-1 when we give

input U then we get state X at t-1 and its

measurement Z at time t-1 .Similarly at time t

we give input and obtain the state X at time t

whose measurement is Z

t .

Figure 3.1 - Markov Assumption

Thus on observing the pattern we see that state X at time t depends upon state X at time t-1 and

input U at time t while the measurement at time t depends on the state X at time t only

.Mathematically it may be represented as :

The final formulation of the Bayes filter is obtained by the combination of the Bayes

theorem and Markov's assumption which is described in detail in Appendix-1.

The final equation obtained is :

From this we observe that Posterior of State = Likelihood of state * (Previous Estimation)

Kalman Filter is for linear systems where - the state

X and the measurement Z can be represented

linearly as-

( w in Z denotes the error in measurement)

3.1.2 Kalman Filter

If we were to track the position and velocity of a moving object then the state may be

written as-

While using one dimensional equation the state and new covariance matrix may be represented

as-

Here B represents the command matrix whereas Q represents the process noise matrix and the

above equation also represents the Prediction step in KF.

Heading on to the Updation step we take the results from measuring devices like

sensors and model them and finally combine with the prediction results by making use of

gaussian multiplication to obtain the final best estimate as :

Here K represents the gain obtained during the gaussian multiplication.The detailed

Mathematical derivation is present in Appendix-2.

Chapter 4

Results and Discussion

Finally the example of a moving body was modeled in Python Code.

The Python Code [4] was referred to and was formulated as :

● Initial state of system was taken in Gaussian form as

Figure-4.1- Initial State

● Then an error of given range was introduced shown by the graph-

Figure-4.2 - Error introduced

● Next the predicted and measured values were obtained.

● Then gaussian multiplication was carried out to obtained the optimum result represented

by the dotted lines in the figure-

Figure-4.3-Predicted , Measured and Optimum state

● Then the results were observed when the belief or K (gain) was varied

Figure-4.4- Variation of the Optimum value as Gain K is changed.

Thus from the above obtained graphs we may conclude that the optimum result varies

with the Predicted and the Measured value and the gain k which represent the trust or

certainty we have in either of the values .

We also see that the variance of optimum result is lesser ( higher precision) than

both the predicted and measured value from figure-4.3.

From figure 4.4 we see that if we have higher certainty in one value be it Predicted or

measured , the optimum graph lies closer to the value having higher certainty.

Chapter 5

Summary and Conclusions

The initial phase began with introduction to the Kalman Filters and their applications

.From there the literature reviews were discussed along with the key takeaways from the

respective papers.

Further the important theoretical concepts and important Mathematical equations

were discussed while finally heading over to the results obtained by the code.

From the above we may conclude that Kalman Filter is an Optimal Estimator for

Linear systems .

Chapter 6

Appendix

6.1 Appendix-1

Derivation of Bayes Filter using Bayes theorem and Markov’s Assumptions:

Probability of area shown in figure 6.1 can be represented as

Figure-6.1

On extending the result -

Now on combining the above result and Markov Assumption we will get-

6.2 Appendix-2

The linear state may be represented as-

Figure-6.2- state multiplied by a scalar K

Now on modeling the belief as a Gaussian distribution-

Figure-6.3-represents gaussian multiplication

6.2.1 Prediction Step

6.2.2 Updation Step

‘Thus the final optimum equation obtained is-

Chapter 7

Literature Cited

1. Kalman filter

https://en.wikipedia.org/wiki/Kalman_filter

2. Constrained Nonlinear State Estimation Using Ensemble Kalman Filters

J. Prakash, Sachin C. Patwardhan, and Sirish L. Shah

Industrial & Engineering Chemistry Research 2010 49 (5), 2242-2253

DOI: 10.1021/ie900197s

3. S. Kolås, B.A. Foss, T.S. Schei,

Constrained nonlinear state estimation based on the UKF approach,

Computers & Chemical Engineering,

Volume 33, Issue 8,2009,

ISSN 0098-1354,

https://doi.org/10.1016/j.compchemeng.2009.01.012.

4.GitHub - rlabbe/Kalman-and-Bayesian-Filters-in-Python

https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python